We then simulated the required SNR over the Gaussian channel with the DVB-S2 low-density parity check code [1]} having a code rate of \(5/6\) , with a maximum number of decoding iterations of 50.
Fig. REF  shows the simulated required SNR with the maximum number of FEC decoding iterations.
The required SNR is defined at an asymmetric information [2]}, [3]}The asymmetric information is equivalent to normalized generalized mutual information [4]}, [5]} and achievable FEC rate [6]} under matched decoding. of \(0.87\) , which corresponds to a pre-FEC BER of about \(3.4\cdot 10^{-2}\) . We observed no residual bit errors in more than \(2.5 \cdot 10^6\)  information bits after FEC decoding in the SNR regime larger than the required SNR for each case.
We simulated both mismatched and matched decoding. The true transmitted two-dimensional symbol PMF is denoted by \(P_{X_{\mathrm {c}}(P_S(1))}\)  and the transmitted symbol PMF assumed in the soft demapping by \(Q_{X_{\mathrm {c}}(P_S(1))}\) . In the matched case, \(Q_{X_{\mathrm {c}}(P_S(1))}\)  was set to \(P_{X_{\mathrm {c}}(P_S(1))}\)  for all \(P_S(1)\) , while in the mismatched case, \(Q_{X_{\mathrm {c}}(P_S(1))}\)  was set to a fixed PMF of \(P_{X_{\mathrm {c}}(0.5)}\)  regardless of the true \(P_S(1)\) .
This is because under dynamically variable source situations, the true transmitted symbol PMF \(P_{X_{\mathrm {c}}(P_S(1))}\)  is hard to track in deployable systems.
As shown in Fig. REF , the required SNR can be reduced by compressed shaping, in contrast to the fixed required SNR by CCDM and uniform QAM with bit scrambling due to the fixed \(P_{X_{\mathrm {c}}}\)  (\(=P_{X_{\mathrm {c}}(0.5)}\) ). The SNR penalty by the mismatch between \(P_{X_{\mathrm {c}}(P_S(1))}\)  and \(Q_{X_{\mathrm {c}}(P_S(1))}\)  is not significant except at very small \(P_S(1)\) .
